GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 3, 1997, 219-222
UPPER ESTIMATE OF THE INTERVAL OF EXISTENCE
OF SOLUTIONS OF A NONLINEAR TIMOSHENKO
EQUATION
DRUMI BAINOV AND EMIL MINCHEV
Abstract. Solutions to the initial-boundary value problem for a nonlinear Timoshenko equation are considered. Conditions on the initial
data and nonlinear term are given so that solutions to the problem
under consideration do not exist for all t > 0. An upper estimate of
the t-interval of the existence of solutions is obtained. An estimate of
the growth rate of the solutions is given.
1. Introduction
Let Ω ⊂ Rn be a bounded domain with boundary ∂Ω and Ω = Ω ∪ ∂Ω.
It is assumed that the divergence theorem can be applied on Ω. Let
š
“1/q
›
’ Z
q
q
L (Ω) = u : kukq,Ω =
< +∞ .
|u(x)| dx
Ω
Denote by u the complex conjugate of u.
We consider the initial-boundary value problem (IBVP) for the nonlinear
Timoshenko equation
utt − ϕ(k∇uk22,Ω )∆u + ∆2 u = |u|p−1 u,
u(0, x) = u0 (x),
x ∈ Ω,
ut (0, x) = u1 (x), x ∈ Ω,
Œ
Œ
= 0, t ≥ 0,
u(t, x)Œ
x∈∂Ω
Œ
Œ
∆u(t, x)Œ
= 0, t ≥ 0,
t ≥ 0, x ∈ Ω,
(1)
(2)
(3)
(4)
(5)
x∈∂Ω
1991 Mathematics Subject Classification. 35B30, 35B35.
Key words and phrases. Nonlinear Timoshenko equation, initial-boundary problem,
interval of existence, estimate of growth rate of solutions.
219
c 1997 Plenum Publishing Corporation
1072-947X/97/0500-0219$12.50/0 
220
DRUMI BAINOV AND EMIL MINCHEV
where p > 1 is a constant, ϕ : R+ → R+ , R+ = [0, +∞), u0 , u1 : Ω → C are
given functions.
Equations of type (1) appear in a variety of models describing nonlinear
vibration of beams. This kind of equations is the object of long-standing
interest. We should mention the papers [1]–[6], as well as the monograph [7].
In the present paper we investigate the blowing up of solutions [8] of the
initial boundary value problem for the nonlinear Timoshenko equation.
2. Main Results
Theorem 1. Let the following conditions hold:
1. u is a suitably smooth classical solution of the IBVP (1)–(5);
Rs
2. ϕ ∈ C(R+ , R+ ) and mΦ(s) ≥ sϕ(s), ∀s ≥ 0, where Φ(s) = ϕ(k) dk,
0
m ≥ 1; R
3. Re u0 u1 dx > 0;
Ω
2
2
+ Φ(k∇u0 k22,Ω ) + k∆u0 k22,Ω − p+1
4. E(0) = ku1 k2,Ω
ku0 kp+1
p+1,Ω ≤ 0;
5. p > 2m − 1.
Then the maximal time interval of the existence [0, Tmax ) of u can be
estimated by
Z∞ ”
Tmax ≤ T0 =
C1 − 4mE(0)ξ +
4C p+3
ξ 2
p+3
2
ku0 k2,Ω
•− 21
dξ < +∞,
where
”
•2
Z
4C
C1 = 2 Re u0 u1 dx + 4mE(0)ku0 k22,Ω −
ku0 kp+3
2,Ω ,
p+3
Ω
’
“ p−1
2
1
p + 1 − 2m
C=2
p+1
mes Ω
and if Tmax = T0 , then
2
= +∞
lim ku(t)k2,Ω
t→T0−
and
p−1
2
lim (T0 − t)ku(t)k2,Ω
−
t→T0
2
≤
p−1
r
p+3
.
C
NONLINEAR TIMOSHENKO EQUATION
221
Proof. We multiply both sides of equation (1) by u, take the real parts of
both sides, and integrate to obtain
’
“
1 d2
2
2
)k∇uk22,Ω +k∆uk22,Ω = kukp+1
kuk2,Ω −kut k22,Ω +ϕ(k∇uk2,Ω
p+1,Ω . (6)
2 dt2
On the other hand, if we multiply both sides of equation (1) by ut , take
the real parts of both sides, and integrate, we get the energy identity
2
2
kut k22,Ω + Φ(k∇uk2,Ω
−
) + k∆uk2,Ω
2
kukp+1
p+1,Ω = E(0).
p+1
(7)
Thus, from (6), (7) and condition 2 of the theorem we conclude that
’
“
1 d2
2
2
2
− k∆uk22,Ω + kukp+1
kuk2,Ω ≥ kut k2,Ω
p+1,Ω + mkut k2,Ω +
2 dt2
2m
2
+ mk∆uk2,Ω
−
kukp+1
p+1,Ω − mE(0) ≥
p+1
p + 1 − 2m
kukp+1
≥ −mE(0) +
p+1,Ω ≥
p+1
’
“ p−1
2
1
p + 1 − 2m
p+1
≥ −mE(0) +
kuk2,Ω
,
p+1
mes Ω
where we have used the Hölder inequality in the last step.
Denote X(t) = ku(t, ·)k22,Ω . Then we have the differential inequality
p+1
d2 X
≥ −2mE(0) + CX 2 ,
dt2
and
X(0) =
ku0 k22,Ω ,
dX
(0) = 2 Re
dt
Z
(8)
u0 u1 dx.
Ω
It is easy to prove that X 0 (t) > 0 wherever X(t) exists. If this conclusion
is false then the set Q = {t : X 0 (t) ≤ 0} is nonempty. Denote t0 = inf Q.
By integrating, we obtain
0
0
0 = X (t0 ) ≥ X (0) − 2mE(0)t0 + C
Zt0
X
p+1
2
(τ ) dτ > 0,
0
which is a contradiction. We multiply both sides of (8) by X 0 (t) and integrating twice we conclude that
Tmax ≤ T0 =
Z∞ ”
ku0 k22,Ω
C1 − 4mE(0)ξ +
4C p+3
ξ 2
p+3
•− 21
dξ < +∞.
222
DRUMI BAINOV AND EMIL MINCHEV
If Tmax = T0 then limt→T − ku(t)k22,Ω = +∞ and from (8) we have
0
T0 − t ≤
Z∞ ”
4C p+3
C1 − 4mE(0)ξ +
ξ 2
p+3
X(t)
≤
r
p+3 4
1
,
4C p − 1 [X(t)] p−1
4
•− 21
dξ ≤
t ∈ [T0 − ε, T0 ),
where ε > 0 is sufficiently small. Thus we obtain
r
p−1
2
p+3
2
lim (T0 − t)ku(t)k2,Ω ≤
.
p−1
C
t→T0−
Remark 1. When ϕ(s) = 1 + s we have m = 2 and p > 3.
Acknowledgements
This investigation was partially supported by the Bulgarian Ministry of
Education, Science and Technologies under Grant MM–422.
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(Received 19.02.1996)
Authors’ address:
Medical University of Sofia
P.O. Box 45, Sofia 1504, Bulgaria